PDE-Constrained optimization problems arise in many different scientific and engineering applications. First, we will present several efficient optimize-then-discretize algorithms for iteratively solving the first-order optimality KKT system from both parabolic and wave PDE-Constrained optimization problems. Second, we will discuss some interesting numerical issues regarding the discretize-then-optimize algorithms, which are also widely used for solving PDE-Constrained optimization problems. Numerical results will be shown to illustrate the effectiveness of our proposed numerical algorithms.